Fejér saturation on a Fourier mode ($\Vert {\sigma }_n(\cos kx)-\cos kx{\Vert }_{\infty }=k/n$)

For fixed positive integer $k$ and $n>k$, the Fourier coefficient of the Fejér kernel at frequency $k$ multiplies the mode by $1-k/n$. The displayed error is exact. Consequently the order $n^{-1}$ cannot be replaced by $o(n^{-1})$ for all twice continuously differentiable periodic functions, even though each fixed Fourier mode is infinitely differentiable.